# How do you differentiate f(x)=4x^5-5x^4 using the sum rule?

Apr 27, 2018

$20 {x}^{4} - 20 {x}^{3}$ or $20 {x}^{3} \left(x - 1\right)$

#### Explanation:

Given: $f \left(x\right) = 4 {x}^{5} - 5 {x}^{4}$.

The sum rule for differentiation states that $\frac{d}{\mathrm{dx}} \left(f \left(x\right) \pm g \left(x\right)\right) = f ' \left(x\right) \pm g ' \left(x\right)$

$\therefore f ' \left(x\right) = \left(4 {x}^{5}\right) ' - \left(5 {x}^{4}\right) '$

Now, we use the power rule, which states that, $\frac{d}{\mathrm{dx}} \left(n {a}^{x}\right) = n x {a}^{x - 1} , x \ne - 1$.

And so, we get:

$f ' \left(x\right) = 5 \cdot 4 {x}^{4} - 4 \cdot 5 {x}^{3}$

$= 20 {x}^{4} - 20 {x}^{3}$

We can factor this into:

$= 20 {x}^{3} \left(x - 1\right)$